Question 483607: If |z|=5, then find maximum value of |z+3+4i|. i need quick answer because i have solved most of the tough questions but confused in solving this one.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! The set of values z satisfying |z| = 5 is simply a circle with radius 5 centered at the origin. It shouldn't be hard to see that z + 3 + 4i is simply that circle, shifted by 3+4i (i.e. 3 units right, 4 units up). To maximize |z + 3 + 4i| think of a triangle ABC with vertices A = 0, B = 3+4i, C is some unknown point on the circle. AB and BC have fixed lengths; however AC can vary, and we want to maximize AC. The maximal value occurs when ABC is a degenerate triangle, i.e. ABC is a straight line. Hence, C is located at 6+8i, and the maximum value of |z+3+4i| is equal to |6+8i| = 10.
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